Optimising Box-Behnken designs for new optimal customised alpha level second order response surface designs

We introduce a new Box-Behnken design style second-order response surface design having ability to fit cubic models called the butterfly design, named for its distinctive geometry. Unlike the existing designs that are effective at identifying optima near centre, the butterfly design locates optima in four bilateral off-centre directions. For its construction, instead of the cubic points, the cuboidal points consisting of factorial-alpha (α) 22 in (±1, ±α) levels with customised α > 1, are inserted at the treatments of special Youden square design, defined in this article. This modification improves accuracy and power of estimation, minimises prediction standard errors while requiring factors to be set at the fixed five levels - the features often compromised in algorithmically optimised designs. The construction provides a new D-, I-optimal 17-run rotatable design in three factors with α = 1:6181, supports augmenting categorical factors, enhances process modelling, and produces run-economical fractional butterfly design, illustrations are provided.